The determination of physical properties of the formations encountered when drilling into the crust of the earth is of great interest in relation to the exploitation of hydrocarbon reservoirs or mining applications.
Several techniques are known for recording physical measurements along a borehole. The results of such techniques are frequently recorded in the form of a so-called well log which provides a record of one or more physical measurements as a function of depth in a well bore.
Such well logs are useful for identifying and correlating underground rocks, as well as for determining the mineralogy and physical properties of potential reservoir rocks and the nature of the fluids they contain. The process of determining the physical properties of the formation based on the measured data is generally referred to as data inversion. In particular, the physical properties may be determined from the measured data and a suitable physical model, for example as model parameters of the physical model which may be determined by fitting the measured data to the model.
One useful technique for determining the elastic properties of a formation includes acoustic measurements. Accordingly seismic inversion refers to the process of determining what physical characteristics of rocks and fluids could have produced a seismic record obtained, e.g. in response to acoustic stimuli generated by one or more acoustic sources at respective positions along the well.
To this end, efficient numerical simulation processes based on computational models of the acoustic properties of the formation are of interest in seismic inversion so as to estimate the physical properties of the formation in an efficient and accurate manner, e.g. to estimate the spatial distribution of elastic formation properties from given borehole acoustic measurements.
Formations encountered when drilling into the crust of the earth may be elastically anisotropic and the axis of anisotropy may change with depth. In order to model the measurements by sonic tools in highly anisotropic formations, and thereby to invert elastic properties from single well data, full 3-dimensional analyses of acoustic wave propagation is desirable.
However, the simulation of borehole acoustic problems has been found to be a difficult task, in particular because the computational simulation model used in the simulation needs to be sufficiently complex and accurate to capture the relevant physical mechanisms while remaining computationally tractable. The finite element method is capable of capturing almost any degree of complexity, e.g. layer by layer variations in elastic parameters and anisotropy, by using standard three dimensional isoparametric elements as described in Klaus-Jurgen Bathe “Finite Element Procedures in Engineering Analysis, Prentice-Hall”. However, the use of such elements for three dimensional modelling becomes computationally exhaustive and prohibitive for simultaneous inversion in logging while drilling applications.